Heat remaining 1 tablespoon vegetable oil in same skillet over medium-high heat. Add tortillas in single layer, draping up sides of skillet to fit. Divide mushroom mixture among tortillas, mounding on only 1 side. Place slice of Monterey Jack cheese atop filling in each tortilla. Fold plain tortilla halves over filling and press firmly. Cook until tortillas are brown, about 1 minute per side. Transfer tacos to plates. Open tacos; sprinkle with chopped cilantro, crumbled feta or Cotija cheese, and toppings.